Alex learned $80$ new vocabulary words for his English exam. The following function gives the number of words he remembers after $t$ days: $W(t)=80(1-0.1t)^3$ What is the instantaneous rate of change of the number of words Alex remembers after $5$ days? Choose 1 answer: Choose 1 answer: (Choice A) A $10$ days (Choice B) B $10$ words per day (Choice C) C $-6$ days (Choice D) D $-6$ words per day
Solution: Understanding the problem The instantaneous rate of change of $W(t)$ is given by its derivative, $W'(t)$. Therefore, the instantaneous rate of change of the number of words Alex remembers after $5$ days is $W'(5)$. Let's find $W'(t)$ and evaluate it at $t=5$. Finding $W'(t)$ $W'(t)=-24(1-0.1t)^2$ Finding $W'(5)$ $\begin{aligned} W'(5)&=-24(1-0.1(5))^2 \\\\ &=-24(0.5)^2 \\\\ &=-6 \end{aligned}$ Interpreting units $W(t)$ is the number of ${\text{words}}$ Alex remembers after $t$ ${\text{days}}$. Therefore, we measure its rate of change in ${\text{words}}$ per ${\text{day}}$. In conclusion, the instantaneous rate of change of the number of words Alex remembers after $5$ days is $-6$ words per day. The rate of change is negative because Alex remembers fewer words every day.